The present invention relates to a method and apparatus for demodulating an electrical signal. The method and apparatus of the present invention are particularly useful for demodulating a differential phase-shift keying signal, and are therefore described below particularly with respect to this application.
In traditional phase modulation, a data signal .phi.(t) is impressed on a carrier signal of frequency f.sub.c to produce a modulated signal of amplitude A EQU A cos(2.pi.f.sub.c t+.phi.(t))
where the phase angle .phi.(t) is either 0 or .pi. and may change only at integral multiples of the symbol (or bit period) duration .tau.. In a conventional method of differential demodulation, a delayed version of the received signal is used for phase reference to the incoming signal, whereupon a multiplier is often used to generate the product: EQU A.sup.2 cos(2.sigma.f.sub.c t+.phi.(t))cos(2.pi.f.sub.c (t-.tau.)+.phi.(t-.tau.))
After lowpass filtering this includes only the phase-difference term: EQU 1/2A.sup.2 cos(2.pi.f.sub.c .tau.+.phi.(t)-.phi.(t-.tau.))
It would be convenient if the term product f.sub.c .tau. would be an integer because then EQU cos(2.pi.f.tau.+.phi.(t)-.phi.(t-.tau.))=cos(.phi.(t)-.phi.(t-.tau.))=cos.D ELTA..phi.
and since the phase difference, which we denote by .DELTA..phi.=.phi.(t)-.phi.(t-.tau.), is either 0 or .pi., depending on the modulating data, the cosine of it is respectively, 1 or -1, yielding exact demodulation. If, however, the term product f.sub.c .tau. is not an integer because of uncertainties of frequency, and if the quantity denoted by .alpha. is equal to 2.pi.f.sub.c .tau., modulo 2.pi., then the following phase shifted angle obtains: EQU cos(.phi.(t)-.phi.(t-.tau.)+.alpha.)=cos.DELTA..phi.cos .alpha.-sin.DELTA..phi.sin .alpha.
Since sin.DELTA..phi.=0 for both signal data alternatives, the result reduces to cos.DELTA..phi.cos .alpha.; and if .alpha. is arbitrary, cos .alpha. can take on arbitrary values between -1 and +1, including zero, in which case the output is useless.
The inevitable conclusion is that under frequency uncertainties, where the term product f.sub.c .tau. can shift from a pre-designed integer value by a fraction (.delta.f).tau. such that 2.pi.(.delta.f).tau. becomes comparable to 1 radian, the conventional detection method is not satisfactory.
Such frequency uncertainty is present in various radio frequency applications because of, for example, unknown Doppler shifts, as well as in optical communication systems that use semiconductor lasers of relatively unstable frequency. It should be noted that .alpha. is in effect a time-varying quantity but its variations, mostly because of the thermal effects in the oscillators and in varying Doppler shifts, are relatively slow, as compared to the data rate in .phi.(t).